My Omnidimensional AP Research: From a Simple Number Pattern to a General Power-Sum Formula

My Omnidimensional AP Research: From a Simple Number Pattern to a General Power-Sum Formula

By Kumaran Kanniappan

My OMNIBAL research explores a question:

If we know exactly how a sequence is generated, can we calculate its powered total without visiting every term?

For arithmetic progressions, we can express that calculation through a midpoint, symmetry, and reusable moment formulas.

In this article, I explain the approach from a simple example to multidimensional arrangements and higher powers, then describe how I would evaluate its practical usefulness.

1. BEGIN WITH A FAMILIAR PATTERN

Consider:

3, 5, 7, 9, 11, 13, 15, 17

Each number increases by two. This is an arithmetic progression, or AP.

Write it as:

a_i = F + h*i, for i = 0,...,X-1

The parameters are:

- F: first value.

- h: common difference.

- X: number of values.

For this sequence:

F = 3, h = 2, X = 8

Its last value is:

L = F + h*(X-1) = 17

Its midpoint is:

M = (F+L)/2 = 10

2. THE MIDPOINT REVEALS THE STRUCTURE

Pair the values from opposite ends:

3 + 17 = 20

5 + 15 = 20

7 + 13 = 20

9 + 11 = 20

The ordinary sum is therefore:

S_1 = X*M = 8 x 10 = 80

Now express each value relative to the midpoint:

10-7, 10-5, 10-3, 10-1, 10+1, 10+3, 10+5, 10+7

This symmetry is the key to higher powers.

3. WHY ODD OFFSET CONTRIBUTIONS DISAPPEAR

For a pair of offsets +u and -u:

(M+u)^2 + (M-u)^2 = 2*M^2 + 2*u^2

The terms involving M*u cancel.

For cubes:

(M+u)^3 + (M-u)^3 = 2*M^3 + 6*M*u^2

For fourth powers:

(M+u)^4 + (M-u)^4 = 2*M^4 + 12*M^2*u^2 + 2*u^4

The binomial expansion always cancels the odd powers of the offset when opposite pairs are added.

This is why the general formula retains only even-indexed binomial coefficients.

4. DEFINE REUSABLE MOMENTS

Define centered position offsets:

j_i = i - (X-1)/2

Then:

a_i = M + h*j_i

The average even powers of these positions are:

mu_(2m)(X) = (1/X) * sum[i=0,...,X-1] j_i^(2m)

The first moment formulas are:

mu_0(X) = 1

mu_2(X) = (X^2-1)/12

mu_4(X) = (3*X^4-10*X^2+7)/240

These depend on the number of positions. The starting value and spacing enter through other parts of the formula.

5. THE GENERAL FORMULA

For a nonnegative integer power p:

S_p = sum[i=0,...,X-1] (F+h*i)^p

The midpoint expansion gives:

S_p = X * sum[m=0,...,floor(p/2)] C(p,2m) * M^(p-2m) * h^(2m) * mu_(2m)(X)

C(p,2m) is a binomial coefficient, and floor(p/2) means p/2 rounded down to an integer.

The formula has floor(p/2)+1 indexed contributions.

For fourth powers, that means three contributions. For fiftieth powers, it means 26.

This count does not depend on the number of AP terms, although the cost of arithmetic still increases with the size of the numbers involved.

6. A COMPLETE FOURTH-POWER EXAMPLE

For our eight values:

X = 8, M = 10, h = 2

The moments are:

mu_2(8) = 21/4

mu_4(8) = 777/16

The fourth-power formula is:

S_4 = X * [M^4 + 6*M^2*h^2*mu_2 + h^4*mu_4]

Substituting:

S_4 = 8 * [10^4 + 6*(10^2)*(2^2)*(21/4) + 2^4*(777/16)]

S_4 = 8 * [10,000 + 12,600 + 777]

S_4 = 187,016

Direct summation confirms the result.

7. CONNECT THE FORMULA TO DIMENSIONS

Suppose an arrangement has D dimensions, with side lengths n_1,n_2,...,n_D.

Its number of positions is:

X = n_1*n_2*...*n_D

If each side has the same length n:

X = n^D

The eight values above could occupy a line of length eight, a 2 x 4 rectangle, or a 2 x 2 x 2 cube.

Their powered sum stays the same because reshaping preserves the values.

This distinction matters: the arrangement supplies the count; the AP rule supplies the values.

A separate formula is needed when values depend on coordinates in a different way.

8. THE FIVE CHOICES BEHIND A CALCULATION

To move from the general framework to a specific result:

1. Choose the number of dimensions.

2. Choose the positions along each dimension.

3. Choose the starting value.

4. Choose the common difference.

5. Choose the nonnegative integer power.

Then calculate the total position count and midpoint, select the required moment formulas, and evaluate the expression.

The framework accommodates any finite dimensional count and any nonnegative integer power. Negative or fractional powers require different treatment.

9. WHERE I SEE PRACTICAL OPPORTUNITIES

Manufacturing: Planned dimensions that increase regularly can produce immediate material totals and higher-order design aggregates.

Energy planning: Structured ramp schedules can be evaluated directly. Actual demand forecasting remains a separate task.

Semiconductor design: Regularly spaced coordinates support analytical geometry moments.

Scientific computing: Selected structured-grid sums can be reduced analytically. For example:

sum[i=1,...,N] sum[j=1,...,N] (i^2+j^2) = 2*N*sum[k=1,...,N] k^2

This works because the coordinate contributions separate.

Software: Predictable polynomial index weights can sometimes replace repeated aggregation loops with direct calculations.

These opportunities need workload-specific validation. A correct formula alone does not establish commercial impact.

10. HOW I DESCRIBE THE RESEARCH CONTRIBUTION

My proposed contribution is the OMNIBAL midpoint-and-moments presentation, connecting dimensional arrangements, progression parameters, and selected powers within a reusable framework.

The underlying mathematics is classical. Earlier literature includes research on power sums of arithmetic progressions:

Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers (2017).

https://arxiv.org/abs/1707.04451

I therefore treat historical novelty as a question to investigate through explicit comparison. A "no one has ever done this before" claim needs stronger evidence than a working example.

11. MY PROPOSED SMART RESEARCH ROADMAP

Within 30 days: Prepare a reproducible explanation covering definitions, derivation, assumptions, and worked examples for powers 0-10.

Within 45 days: Validate at least 1,000 cases using direct summation and an independent reference method. Document every discrepancy and resolution.

Within 60 days: Benchmark at least five sequence sizes against efficient existing approaches, reporting arithmetic precision, runtime, memory, and preprocessing.

Within 75 days: Compare the framework with earlier AP power-sum methods and clearly state which elements are established and which contribution is being proposed.

Within 90 days: Publish a reproducible demonstration for one industry workload, including its benefits, limitations, and adoption criteria.

These are proposed milestones, not completed achievements.

My aim is to make this research understandable enough to learn, precise enough to verify, and practical enough to test.

The next step is to choose one real calculation and demonstrate exactly what the structure allows us to achieve.